3.25 \(\int \frac{1-b x^2}{\sqrt{-1+b^2 x^4}} \, dx\)

Optimal. Leaf size=89 \[ \frac{2 \sqrt{1-b^2 x^4} F\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{b^2 x^4-1}}-\frac{\sqrt{1-b^2 x^4} E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{b^2 x^4-1}} \]

[Out]

-((Sqrt[1 - b^2*x^4]*EllipticE[ArcSin[Sqrt[b]*x], -1])/(Sqrt[b]*Sqrt[-1 + b^2*x^
4])) + (2*Sqrt[1 - b^2*x^4]*EllipticF[ArcSin[Sqrt[b]*x], -1])/(Sqrt[b]*Sqrt[-1 +
b^2*x^4])

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Rubi [A]  time = 0.150733, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{2 \sqrt{1-b^2 x^4} F\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{b^2 x^4-1}}-\frac{\sqrt{1-b^2 x^4} E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{b^2 x^4-1}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - b*x^2)/Sqrt[-1 + b^2*x^4],x]

[Out]

-((Sqrt[1 - b^2*x^4]*EllipticE[ArcSin[Sqrt[b]*x], -1])/(Sqrt[b]*Sqrt[-1 + b^2*x^
4])) + (2*Sqrt[1 - b^2*x^4]*EllipticF[ArcSin[Sqrt[b]*x], -1])/(Sqrt[b]*Sqrt[-1 +
b^2*x^4])

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Rubi in Sympy [A]  time = 32.2685, size = 82, normalized size = 0.92 \[ - \frac{\sqrt{- b^{2} x^{4} + 1} E\left (\operatorname{asin}{\left (\sqrt{b} x \right )}\middle | -1\right )}{\sqrt{b} \sqrt{b^{2} x^{4} - 1}} + \frac{2 \sqrt{- b^{2} x^{4} + 1} F\left (\operatorname{asin}{\left (\sqrt{b} x \right )}\middle | -1\right )}{\sqrt{b} \sqrt{b^{2} x^{4} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+1)/(b**2*x**4-1)**(1/2),x)

[Out]

-sqrt(-b**2*x**4 + 1)*elliptic_e(asin(sqrt(b)*x), -1)/(sqrt(b)*sqrt(b**2*x**4 -
1)) + 2*sqrt(-b**2*x**4 + 1)*elliptic_f(asin(sqrt(b)*x), -1)/(sqrt(b)*sqrt(b**2*
x**4 - 1))

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Mathematica [C]  time = 0.0499081, size = 73, normalized size = 0.82 \[ \frac{i \sqrt{1-b^2 x^4} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-b} x\right )\right |-1\right )-2 F\left (\left .i \sinh ^{-1}\left (\sqrt{-b} x\right )\right |-1\right )\right )}{\sqrt{-b} \sqrt{b^2 x^4-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - b*x^2)/Sqrt[-1 + b^2*x^4],x]

[Out]

(I*Sqrt[1 - b^2*x^4]*(EllipticE[I*ArcSinh[Sqrt[-b]*x], -1] - 2*EllipticF[I*ArcSi
nh[Sqrt[-b]*x], -1]))/(Sqrt[-b]*Sqrt[-1 + b^2*x^4])

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Maple [A]  time = 0.008, size = 108, normalized size = 1.2 \[ -{1\sqrt{b{x}^{2}+1}\sqrt{-b{x}^{2}+1} \left ({\it EllipticF} \left ( x\sqrt{-b},i \right ) -{\it EllipticE} \left ( x\sqrt{-b},i \right ) \right ){\frac{1}{\sqrt{-b}}}{\frac{1}{\sqrt{{b}^{2}{x}^{4}-1}}}}+{1\sqrt{b{x}^{2}+1}\sqrt{-b{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{-b},i \right ){\frac{1}{\sqrt{-b}}}{\frac{1}{\sqrt{{b}^{2}{x}^{4}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+1)/(b^2*x^4-1)^(1/2),x)

[Out]

-1/(-b)^(1/2)*(b*x^2+1)^(1/2)*(-b*x^2+1)^(1/2)/(b^2*x^4-1)^(1/2)*(EllipticF(x*(-
b)^(1/2),I)-EllipticE(x*(-b)^(1/2),I))+1/(-b)^(1/2)*(b*x^2+1)^(1/2)*(-b*x^2+1)^(
1/2)/(b^2*x^4-1)^(1/2)*EllipticF(x*(-b)^(1/2),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{b x^{2} - 1}{\sqrt{b^{2} x^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x^2 - 1)/sqrt(b^2*x^4 - 1),x, algorithm="maxima")

[Out]

-integrate((b*x^2 - 1)/sqrt(b^2*x^4 - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{b x^{2} - 1}{\sqrt{b^{2} x^{4} - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x^2 - 1)/sqrt(b^2*x^4 - 1),x, algorithm="fricas")

[Out]

integral(-(b*x^2 - 1)/sqrt(b^2*x^4 - 1), x)

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Sympy [A]  time = 3.94169, size = 60, normalized size = 0.67 \[ \frac{i b x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{b^{2} x^{4}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} - \frac{i x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{b^{2} x^{4}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+1)/(b**2*x**4-1)**(1/2),x)

[Out]

I*b*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), b**2*x**4)/(4*gamma(7/4)) - I*x*ga
mma(1/4)*hyper((1/4, 1/2), (5/4,), b**2*x**4)/(4*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{b x^{2} - 1}{\sqrt{b^{2} x^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x^2 - 1)/sqrt(b^2*x^4 - 1),x, algorithm="giac")

[Out]

integrate(-(b*x^2 - 1)/sqrt(b^2*x^4 - 1), x)